A capped call option has the payoff at the expiry T below:

min$($max$($S $$ K$,0),$ H$)$

where H $>=0$ is a given cap. Similarly a capped put has the payoff

min$($max$($K $$ S$,0),$ H $).$

Assume the stock price follows an N$-$period binomial model with u $=$ e$\backslash$sigma $$t and d $=$ e$\backslash$sigma $$t$.$ Write the MATLAB or Python scripts to compute European capped put$/$call option values by using the data in Table $1.$ Using $$t $=0\mathrm{.}0025$ to construct a multi$-$period binomial model with N $=$ T$/$t to compute the initial values of European capped puts with the expiry T $=1,$ H $=10,$ and the strike K $=60$ : $10$ : $100.$ Plot the initial value against the strike. How does the capped put value compare to that of a standard European put with the same strike K and expiry T$?$ Comment and explain your observations. In particular, your codes should look like: $($Hint: the risk$-$neutral probability for up$,$ p $=$ e$^($r$$t$)$e$^(\backslash$sigma $*$sqrt$($t$))/$e$^($sigma$*$sqrt$($delta$*$t$))-$e$^(-$sigma$*$sqrt$($delta$*$t$))$ ; the probability for down is q$=1-$p$.$ Use the continuous interest compound.$)$

$[$Table $1.$ Some typical option parameters

$\backslash$sigma $=30\%$

r$=1\mathrm{.}5\%$

Time to expiry $=1\mathrm{.}0$ years Initial asset price S$0=$$$100][1\%$This function evaluates the arbitrage$-$free price of a European capped put $2\%$ option in the Binomial tree model

$3\%$ INPUTS

$4\%$S$0$:stockpriceattime$0$

$5\%$ sigma : the volatility of stock $6\%$dt :Deltat

$7\%$ T : number of periods

$8\%$ r : interest rate

$9\%$K :strikepriceoftheput

$10\%$H :capofthepput

$11\%$ OUTPUT $]$

$12\%$y :priceoftheputattimezero